Multiplicative functions on shifted primes

Let $f$ be a positive multiplicative function and let $k\geq 2$ an integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently slowly as $p\rightarrow +\infty$, in sense $\sum_{p}|f(p)-1|=\infty$, there exists real number $c>0$ such $k$-tuples $(f(p+1),\ldots,f(p+k))$ are dense hypercube $[0,c]^k$ or $[c,+\infty)^k$. In particular, $f(p+1),\ldots,f(p+k)$ can put any increasing order infinitely often. Our work generalises previous results of De Koninck Luca.